Abstract Erdős proved that F ( A ) : = ∑ a ∈ A 1 a log ⁡ a converges for any primitive set of integers A and later conjectured this sum is maximized when A is the set of primes. Banks and Martin further conjectured that F ( P 1 ) > ⋯ > F ( P k ) > F ( P k + 1 ) > ⋯ , where P j is the set of integers with j prime factors counting multiplicity, though this was recently disproven by Lichtman. We consider the corresponding problems over the function field F q [ x ] , investigating the sum F ( A ) : = ∑ f ∈ A 1 deg f ⋅ q deg f . We establish a uniform bound for F ( A ) over all primitive sets of polynomials A ⊂ F q [ x ] and conjecture that it is maximized by the set of monic irreducible polynomials. We find that the analogue of the Banks-Martin conjecture is false for q = 2 , 3, and 4, but we find computational evidence that it holds for q > 4 .

中文翻译：

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关于函数域中的 Erdős 本原集猜想

摘要 Erdős 证明了 F ( A ) : = ∑ a ∈ A 1 a log ⁡ a 收敛于任何原始整数集 A 并且后来推测当 A 是素数集时该和最大化。Banks 和 Martin 进一步推测 F ( P 1 ) > ⋯ > F ( P k ) > F ( P k + 1 ) > ⋯ ，其中 P j 是具有 j 个素因数的整数集合，尽管这最近被证明是错误的通过利希特曼。我们考虑函数域 F q [ x ] 上的相应问题，研究和 F ( A ) : = ∑ f ∈ A 1 deg f ⋅ q deg f 。我们在多项式 A ⊂ F q [ x ] 的所有原始集合上为 F ( A ) 建立了一个统一的界限，并推测它是由一组不可约多项式的 monic 最大化的。我们发现 Banks-Martin 猜想的类比对于 q = 2 、 3 和 4 是错误的，